It's not "forcing" anyone. People can have as many kids as they want....they would just need to pay for the care of any children beyond the first 2.
I would love to hear how "we" can help stop this pattern that has helped ruin our healthcare system.
You'd probably need to do it from the opposite end. Instead of using an ultimatum, you would need to incentivize having the procedure.
For example, after a women gives birth (because that's the easiest time to do the procedure) offer a monetary incentive, perhaps in the form of a reimbursement to have a tubal performed.
Telling a woman that she either needs to have her tubes tied or lose her welfare would obviously never fly.
For men, offer the surgery for free with a similar incentive.